The main goal of this paper is to develop
first- and second-order energy dissipative and MBP-preserving SAV-type schemes for the Allen–Cahn type equation (REF )
by using an appropriate stabilization technique.
Specifically,
we propose new stabilized exponential-SAV (ESAV) schemes
by introducing an artificial stabilization term rather than basing on the splitting of the energy functional suggested in [1]}.
With the effect of such stabilization, we show that
the proposed first-order scheme preserves the MBP unconditionally with an appropriate stabilizing parameter
and the second-order one does under a time step size constraint.
A main difficulty for numerical analysis of the two schemes lies in that
the coefficients of the nonlinear term and the stabilization term are varying rather than constant
due to the use of the SAV approach.
With the help of the energy dissipation and MBP, we are able to show that such variable coefficients are bounded from above and below by certain positive constants, and consequently, optimal error estimate are successfully obtained for the proposed schemes.
To the best of our knowledge, this is the first work in the direction of designing such SAV-type methods. More importantly, the proposed stabilizing approaches can be easily generalized to deal with many other type of gradient flow problems where the SAV methods apply.
